Overview

The shear modulus, often denoted G and also called the modulus of rigidity, measures how resistant a material is to deformation under shear stress or twisting. Unlike bulk modulus, which describes volumetric response, the shear modulus describes change of shape (distortion) at roughly constant volume. It is a fundamental elastic property used in structural design, seismology and materials characterization.

Definition and units

In linear elasticity the shear modulus is defined as the ratio of shear stress to shear strain: τ = G·γ, where τ (tau) is the applied shear stress and γ (gamma) is the resulting angular shear strain. The SI unit of G is the pascal (Pa), equivalent to newtons per square metre. The same material property is sometimes symbolized by μ in continuum mechanics.

Measurement and examples

Common experimental methods to determine G include torsion tests on cylindrical rods, shear rheometry for soft materials, and ultrasonic or dynamic mechanical analysis for small-strain properties. Typical examples: metals and ceramics have relatively high shear moduli, polymers and rubbers show much lower static shear moduli and often strong frequency- or temperature-dependence, and fluids have negligible static shear modulus but can display a complex shear response if viscoelastic.

  • Torsion of a shaft: torque relates to G and the shaft's polar moment of inertia.
  • Seismology: shear-wave speed through rock depends on G and density.
  • Rheology: dynamic (complex) shear modulus characterizes viscoelastic materials.

Relations and distinctions

For isotropic linear elastic materials the shear modulus is related to Young's modulus E and Poisson's ratio ν by G = E / [2(1+ν)]. This relation does not hold for anisotropic materials, where multiple shear moduli may be required to describe different shear planes and directions. Shear modulus differs conceptually from bulk modulus (resistance to uniform compression) and from Young's modulus (axial stiffness).

Notable facts and applications

Engineers use G to predict torsional stiffness and shear stresses in shafts, fasteners and beams. Geophysicists derive shear modulus from seismic shear-wave velocities to infer subsurface rigidity. In polymers and biological tissues the shear response often requires a complex-valued G(ω) to capture time-dependent (viscoelastic) behavior. For further reading see shear force resources and shear strain explanations.